Chapter 3 – Time value of Money

This chapter discusses how to calculate the present value, future value, internal rate of return, and modified internal rate of return of a cash flow stream. Understanding how (and when) to use these formulas is essential to your success as a financial manager! Formulas and examples are included with these notes.

Numbers are rounded to 4 decimal places in tables and formula. However, the actual (non-rounded) numbers are used in the calculations.
Time Value of Money Concepts

Financial Calculator: N = 1, I/Y = 5, PV = -1, PMT = 0, FV = Answer
Note on financial calculators – The calculator inputs described above are for a Texas Instruments BAII Plus calculator. (Many other financial calculators require similar inputs.)
Notice that you enter a -1 as the PV and the solution is +1.05. Here is the intuition: deposit $1 in the bank (negative cash flow), withdraw $1.05 in one year (positive cash flow). If you had entered +1 as the PV, the solution would be –1.05.

Future value of $1, as of time 5. Interest rate = 5%.

0

1

2

3

4

5

$1

$0

$0

$0

$0

$0

Formula: C0 (1 + r)t = $1(1.05)5 = $1.2763

Financial Calculator: N = 5, I/Y = 5, PV = -1, PMT = 0, FV = Answer

Present value of $1, received at time 1. Discount rate = 5%.

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2

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5

$0

$1

$0

$0

$0

$0

Formula: C1 / (1 + r)t = $1/(1.05)1 = $0.9524

Financial Calculator: N = 1, I/Y = 5, PV = Answer, PMT = 0, FV = -1

Present value of $1, received at time 5. Discount rate = 5%.

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5

$0

$0

$0

$0

$0

$1

Formula: C5 / (1 + r)t = $1/(1.05)5 = $0.7835

Financial Calculator: N = 5, I/Y = 5, PV = Answer, PMT = 0, FV = -1

Compounding periods less than one year

Future value of $1, as of time 5. Interest rate = 5%, compounded “m” times per year.

0

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5

$1

$0

$0

$0

$0

$0

General (non-continuous) formula: C0 (1 + r/m)tm

Continuous compounding formula: C0 ert

Note: “e” = 2.718281828

Semi-annual compounding: $1(1 + (0.05/2))(5)(2) = $1.280085

Monthly compounding: $1(1 + (0.05/12))(5)(12) = $1.283359

Daily compounding: $1(1 + (0.05/365))(5)(365) = $1.284003

Continuous compounding: $1 e(5)(0.05) = $1.284025

Note: Some may use 360 days as the length of one year, other may take into account leap years (366 days every four years). The effects of these changes (from a 365-day year) are extremely small.
Financial Calculator (for semi-annual): N = 10, I/Y = 5/2, PV = -1, PMT = 0, FV = Answer

This standard formula for the future value of a finite annuity gives a value as of the last period of the annuity (time 4 in this example). The 1.05 is raised to the fourth power because there are 4 payments in the annuity.

Value as of time 4 = $1 [(1.054 – 1) / 0.05] =

$4.3101

You can calculate the value of the cash flows at other points in time by multiplying or dividing by 1+r, where r (the interest and discount rate) is 5% in this example.

For instance, assume you want to know the value of the above cash flow stream at t = 6. Time 6 is two years after time 4. To calculate, use the standard formula to determine the value at t = 4, then multiply by 1.052 to determine the value at t = 6. (Use the second power because you are calculating the value two years after time 4.) The solution is:

Value as of time 6 = $1 [(1.054 – 1) / 0.05] 1.052 =

$4.7519

As a second example, assume that you want to know the value of the above cash flow stream at t = 1. Time 1 is three years before time 4. To calculate, use the standard formula to determine the value at t = 4, then divide by 1.053 to determine the value at t = 1. (Use the third power because you are calculating the value three years before time 4.) The solution is:

Value as of time 1 = $1 [(1.054 – 1) / 0.05] / 1.053 =

$3.7232

Therefore, “multiply” when you want to determine the value at a later date, “divide” when you want to determine the value at an earlier date.

Financial Calculator (time 7): N = 4, I/Y = 5, PV = 0, PMT = -1, FV = Answer. Multiply answer by 1.053You can also use the formula for the present value of a finite annuity to calculate the value of a cash flow stream at different points in time.
Standard formula for the present value of a finite annuity = C { [1 – (1 / (1 + r))t] / r}
The standard formula gives a value one period before the first payment of the annuity (time 0 in this example). The 1.05 is raised to the fourth power because there are 4 payments in the annuity.

Value as of time 0 = $1 { [1 – (1/1.05)4] / (0.05) } =

$3.5460

As before, you can calculate the value at other points in time by multiplying or dividing by 1+r, (1.05 in this example).

Three-year annuity of $1 per year, first cash flow received at t = 0. Interest and discount rate = 5%.

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$1

$1

$1

$0

$0

$0

The standard future value annuity formula gives a value as of the last year of the annuity (year 2 in this example). This is a three-year annuity. Therefore, 1.05 is raised to the third power in the formula.

Value as of time 2 = $1 [(1.053 – 1) / 0.05] =

$3.1525

The value at other points in time can be calculated by multiplying or dividing by 1.05, raised to the appropriate power.

Value as of time 0 = $1 [(1.053 – 1) / 0.05] / 1.052 =

$2.8594

Value as of time 4 = $1 [(1.053 – 1) / 0.05] 1.052 =

$3.4756

The standard present value annuity formula gives a value one period before the first payment of the annuity. Therefore, the formula will give you a value at t = -1. You need to multiply by 1 + r to get the value by t = 0.

Value as of time 0 = $1 { [1 – (1/1.05)3] / (0.05) } 1.051 =

$2.8594

The values at time 2 and 4:

Value as of time 2 = $1 { [1 – (1/1.05)3] / (0.05) } 1.053 =

$3.1525

Value as of time 4 = $1 { [1 – (1/1.05)3] / (0.05) } 1.055 =

$3.4756

Five-year annuity of $1 per year, first cash flow received at t = 3. Interest and discount rate = 5%.

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5

6

7

8

$0

$0

$0

$1

$1

$1

$1

$1

$0

The standard future value annuity formula gives the value as of the last year of the annuity (t = 7 in this example). This is a five-year annuity. Therefore, 1.05 is raised to the fifth power.

Value as of time 7 = $1 [(1.055 – 1) / 0.05] =

$5.5256

Values at different points in time using the future value annuity formula. A couple of examples

Value as of time 0 = $1 [(1.055 – 1) / 0.05] / 1.057 =

$3.9270

Value as of time 4 = $1 [(1.055 – 1) / 0.05] / 1.053 =

$4.7732

Value as of time 8 = $1 [(1.055 – 1) / 0.05] 1.051 =

$5.8019

The standard present value annuity formula gives the value as of the year before the first payment of the annuity (t = 2 in this example).

Value as of time 2 = $1 { [1 – (1/1.05)5] / (0.05) } =

$4.3295

Values at different points in time using the present value annuity formula. A couple of examples:

Standard formula for the present value of a finite growing annuity (for when r is not equal to g) =

Cfirst [1 – [(1 + g) / (1 + r)]t ] / (r – g). This formula gives the value one period before the first payment (t = 0 in this example).
Cfirst is the first cash flow of the annuity. In this above example, Cfirst = C1 = $1.

Value as of time 0 = $1 { [1 – (1.10/1.05)4] / (0.05 – 0.10) }

$4.0904

Values at different points in time using the present value growing annuity formula. Multiply or divide by 1+r (raised to the appropriate power) to determine the value at other points in time.

The standard formula for the present value of a perpetual constant annuity gives you the value one period before the first payment (t = -1 in this example). Therefore, you need to multiply by 1.05 to get the value at t = 0.